So I am currently in the z-transforms unit of my DSP course in school and I'm not sure how to sketch the frequency response of a particular question.

Question: A transfer function H(z), has one zero at the origin of the complex z-plane, another zero at (√2)/2 on the real axis, and a pair of complex conjugate poles at (√2)/2 + j(√2)/2.

a) Draw a pole-zero plot of the transfer function.
b) Without matlab, sketch the magnitude vs. frequency response of the system.
c) Write the transfer function, H(z).
d) Write the discrete-time function, h[n].
e) Write the difference equation for the impulse of the system.
f) Given a sampling frequency of 16kHz, calculate the frequency of oscillation for the system.

The only part I am unsure of is part b). I was able to solve the other questions and I included some solved information in the attached image:

http://img8.imageshack.us/i/ztransformhelp.jpg/

How can I solve for the approximate magnitudes of poles or zeros so I can roughly sketch the response?

 

I don't think I can be of much help, except from giving you the bode plot as produced by Matlab for sampling frequencies of 1Hz (normalized) and 16000Hz.

It seems I need to revise my control systems theoretical basis.

 

One thing I am familiar with is that usually for magnitude plots, we can take 20log(H(z)) substiting z with certain points. However, I'm not sure what we susbtitute z for. I know that in the s-plane, you can find out the magnitude at certain points because for example, τs + 1 = jω/ωc +1. However I do not think that we have learned enough material to directly see how this releates to complex z.

 

Check this out:
http://cnx.org/content/m10548/latest/

Short answer is that you can traverse the unit circle from w = 0 to w = pi and for poles, you amplify and zeros you drop gain. That's just how to do a very rough sketch of the magnitude. You can plug in some values and calculate the magnitude at given frequencies once you have the transfer function.

 

Check this out:
http://cnx.org/content/m10548/latest/

Short answer is that you can traverse the unit circle from w = 0 to w = pi and for poles, you amplify and zeros you drop gain. That's just how to do a very rough sketch of the magnitude. You can plug in some values and calculate the magnitude at given frequencies once you have the transfer function.

I've tried loading the link on two different computers and even after installing the add on, it doesn't load properly.

 

Oh i don't think it works for me either haha. Sorry about that, I more meant the article. It does a good job of describing it!

 

I got the article to load on a third computer haha. After reading the article, it basically confirmed what I know on the rough estimation how steep/shallow/etc. a pole or zero would look like. I do not understand how it tells us how to solve for the approximate magnitude of each pole in dB though.

 

Can you elaborate your question a bit more... i'm not sure I understand.

My interpretation: You know the location of the poles... magnitude of pole = sqrt(RE{p}^2+IM{p}^2)

 

When I entered the information in matlab, the frequency responce I got showed a sharp peak of about 74 dB at the normalized frquency of pi/4. How do you calculate that 74 dB?